Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

Force between magn

The force between two small magnets is quite complicated and depends on the strength and orientation of
both magnets and the distance and direction of the magnets relative to
each other. The force is particularly sensitive to rotations of the
magnets due to magnetic torque. The force on each magnet depends on its
magnetic moment and the magnetic field of the other.

To understand the force between magnets, it is useful to examine the magnetic pole model given above. In this model, the H-field of one magnet pushes and pulls on both poles of a second magnet. If this H-field is the same at both poles of
the second magnet then there is no net force on that magnet since the
force is opposite for opposite poles. If, however, the magnetic field of
the first magnet is nonuniform (such as the H near one of
its poles), each pole of the second magnet sees a different field and
is subject to a different force. This difference in the two forces moves
the magnet in the direction of increasing magnetic field and may also
cause a net torque.

This is a specific example of a general rule that magnets are
attracted (or repulsed depending on the orientation of the magnet) into
regions of higher magnetic field. Any non-uniform magnetic field whether
caused by permanent magnets or by electric currents will exert a force
on a small magnet in this way.

The details of the Amperian loop model are different and more
complicated but yield the same result: that magnetic dipoles are
attracted/repelled into regions of higher magnetic field.
Mathematically, the force on a small magnet having a magnetic moment m due to a magnetic field B is:

where the gradient∇ is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B. To understand this equation, note that the dot productm · B = mBcos(θ), where m and B represent the magnitude of the m and B vectors and θ is the angle between them. If m is in the same direction as B then the dot product is positive and the gradient points 'uphill'
pulling the magnet into regions of higher B-field (more strictly larger m · B).
This equation is strictly only valid for magnets of zero size, but is
often a good approximation for not too large magnets. The magnetic force
on larger magnets is determined by dividing them into smaller regions
having their own m then summing up the forces on each of these regions.